Orders of Vanishing and U(1) Charges in F-theory
arXiv:2110.10159 · doi:10.1007/JHEP03(2022)051
Abstract
Many interesting questions about F-theory models, including several concerning the F-theory swampland, involve massless matter charged under U(1) gauge symmetries. It is therefore important to better understand the geometric properties of F-theory models realizing various U(1) charges. We propose that, for F-theory models described by elliptic fibrations in Weierstrass form, the U(1) charge of light matter is encoded in the orders of vanishing of the section components corresponding to the U(1) gauge symmetry. We give specific equations relating the U(1) charges to the orders of vanishing that seem to hold for both U(1)-charged singlets and for matter additionally charged under a simply-laced nonabelian gauge algebra. Our formulas correctly describe properties of F-theory models in the prior literature, and we give an argument that they should describe the orders of vanishing for arbitrarily high U(1) charges. They also resemble formulas for the -adic valuations of elliptic divisibility sequences developed by Stange [arXiv:1108.3051]. These proposals could serve as a U(1) analogue of the Katz-Vafa method, allowing one to determine U(1) charges without resolution. Additionally, they predict geometric information about F-theory models with general U(1) charges, which may be useful for exploring the F-theory landscape and swampland.
arXiv comment112 pages with appendices; v2: references added
References in corpus (74)
- Evidence for F-Theory
- The Physics of Heavy Z' Gauge Bosons
- Compactifications of F-Theory on Calabi--Yau Threefolds -- II
- Compactifications of F-Theory on Calabi--Yau Threefolds -- I
- Geometric Singularities and Enhanced Gauge Symmetries
- Les Houches Lectures on Constructing String Vacua
- Matter From Geometry
- T-Branes and Monodromy
- F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds
- F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches
- On Abelian Gauge Symmetries and Proton Decay in Global F-theory GUTs
- Classifying bases for 6D F-theory models
- Anomaly Cancellation in Six Dimensions
- Global aspects of the space of 6D N = 1 supergravities
- Matter and singularities
- Anomaly Equations and Intersection Theory
- Tate's algorithm and F-theory
- F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections
- Generalized Green-Schwarz mechanism in F theory
- Chiral Four-Dimensional F-Theory Compactifications With SU(5) and Multiple U(1)-Factors
- Geometric Engineering in Toric F-Theory and GUTs with U(1) Gauge Factors
- Anomaly Cancellation And Abelian Gauge Symmetries In F-theory
- Top Down Approach to 6D SCFTs
- SU(5) Tops with Multiple U(1)s in F-theory
- U(1) symmetries in F-theory GUTs with multiple sections
- T-Branes and Geometry
- Elliptic Fibrations with Rank Three Mordell-Weil Group: F-theory with U(1) x U(1) x U(1) Gauge Symmetry
- Elliptic fibrations for SU(5) x U(1) x U(1) F-theory vacua
- Constraints on 6D Supergravity Theories with Abelian Gauge Symmetry
- TASI Lectures on Supergravity and String Vacua in Various Dimensions
- F-theory GUTs with U(1) Symmetries: Generalities and Survey
- The Global Gauge Group Structure of F-theory Compactification with U(1)s
- Matter From Geometry Without Resolution
- Swampland Bounds on the Abelian Gauge Sector
- Quantization of anomaly coefficients in 6D supergravity
- General U(1)xU(1) F-theory Compactifications and Beyond: Geometry of unHiggsings and novel Matter Structure
- The fate of U(1)'s at strong coupling in F-theory
- F-theory and All Things Rational: Surveying U(1) Symmetries with Rational Sections
- Finite-Dimensional Lie Algebras and Their Representations for Unified Model Building
- 6d SCFTs and U(1) Flavour Symmetries
- A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations
- TASI Lectures on F-theory
- Tate Trees for Elliptic Fibrations with Rank one Mordell-Weil group
- General Prescription for Global U(1)'s in 6D SCFTs
- Tall sections from non-minimal transformations
- F-GUTs with Mordell-Weil U(1)'s
- Origin of Abelian Gauge Symmetries in Heterotic/F-theory Duality
- The Arithmetic of Elliptic Fibrations in Gauge Theories on a Circle
- Rational F-Theory GUTs without exotics
- Generic matter representations in 6D supergravity theories
- Non-Higgsable abelian gauge symmetry and F-theory on fiber products of rational elliptic surfaces
- Exotic matter on singular divisors in F-theory
- Non-Abelian Gauge Symmetry and the Higgs Mechanism in F-theory
- Tate's Algorithm for F-theory GUTs with two U(1)s
- Large U(1) charges in F-theory
- Three-Index Symmetric Matter Representations of SU(2) in F-Theory from Non-Tate Form Weierstrass Models
- On G-flux, M5 instantons, and U(1)s in F-theory
- Abelian F-theory Models with Charge-3 and Charge-4 Matter
- Tuned and Non-Higgsable U(1)s in F-theory
- F-Theory on Quotients of Elliptic Calabi-Yau Threefolds
- High electric charges in M-theory from quiver varieties
- The Toric SO(10) F-Theory Landscape
- F-theory models on K3 surfaces with various Mordell-Weil ranks -constructions that use quadratic base change of rational elliptic surfaces
- General F-theory models with tuned symmetry
- When Rational Sections Become Cyclic: Gauge Enhancement in F-theory via Mordell--Weil Torsion
- Automatic Enhancement in 6D Supergravity and F-theory Models
- Integral points on elliptic curves and explicit valuations of division polynomials
- Global Gluing and G-flux
- F-theory models with 3 to 8 U(1) factors on K3 surfaces
- F-theory models with and transitions in discrete gauge groups
- On genus one fibered Calabi-Yau threefolds with 5-sections
- Half-hypermultiplets and incomplete/complete resolutions in F-theory
- Calabi-Yau 3-folds, Calabi-Yau 3-folds as double covers, and F-theory with U(1)s
- Matter representations from geometry: under the spell of Dynkin